Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  erdszelem10 Structured version   Unicode version

Theorem erdszelem10 28476
Description: Lemma for erdsze 28478. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n  |-  ( ph  ->  N  e.  NN )
erdsze.f  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
erdszelem.i  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.j  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.t  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
erdszelem.r  |-  ( ph  ->  R  e.  NN )
erdszelem.s  |-  ( ph  ->  S  e.  NN )
erdszelem.m  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
Assertion
Ref Expression
erdszelem10  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
Distinct variable groups:    x, y    m, n, x, y, F   
n, I, x, y   
n, J, x, y    R, m, x, y    m, N, n, x, y    ph, m, n, x, y    S, m, x, y    T, m
Allowed substitution hints:    R( n)    S( n)    T( x, y, n)    I( m)    J( m)

Proof of Theorem erdszelem10
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 fzfi 12062 . . . . . . . 8  |-  ( 1 ... ( R  - 
1 ) )  e. 
Fin
2 fzfi 12062 . . . . . . . 8  |-  ( 1 ... ( S  - 
1 ) )  e. 
Fin
3 xpfi 7803 . . . . . . . 8  |-  ( ( ( 1 ... ( R  -  1 ) )  e.  Fin  /\  ( 1 ... ( S  -  1 ) )  e.  Fin )  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  e.  Fin )
41, 2, 3mp2an 672 . . . . . . 7  |-  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  e. 
Fin
5 ssdomg 7573 . . . . . . 7  |-  ( ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  e.  Fin  ->  ( ran  T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  ran  T  ~<_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
64, 5ax-mp 5 . . . . . 6  |-  ( ran 
T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  ran  T  ~<_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
7 domnsym 7655 . . . . . 6  |-  ( ran 
T  ~<_  ( ( 1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  -.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
86, 7syl 16 . . . . 5  |-  ( ran 
T  C_  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ->  -.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
9 erdszelem.m . . . . . . . 8  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  <  N )
10 hashxp 12472 . . . . . . . . . 10  |-  ( ( ( 1 ... ( R  -  1 ) )  e.  Fin  /\  ( 1 ... ( S  -  1 ) )  e.  Fin )  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  =  ( (
# `  ( 1 ... ( R  -  1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) ) )
111, 2, 10mp2an 672 . . . . . . . . 9  |-  ( # `  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )  =  ( (
# `  ( 1 ... ( R  -  1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) )
12 erdszelem.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
13 nnm1nn0 10849 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  ( R  -  1 )  e.  NN0 )
14 hashfz1 12399 . . . . . . . . . . 11  |-  ( ( R  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( R  -  1 ) ) )  =  ( R  -  1 ) )
1512, 13, 143syl 20 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( R  - 
1 ) ) )  =  ( R  - 
1 ) )
16 erdszelem.s . . . . . . . . . . 11  |-  ( ph  ->  S  e.  NN )
17 nnm1nn0 10849 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
18 hashfz1 12399 . . . . . . . . . . 11  |-  ( ( S  -  1 )  e.  NN0  ->  ( # `  ( 1 ... ( S  -  1 ) ) )  =  ( S  -  1 ) )
1916, 17, 183syl 20 . . . . . . . . . 10  |-  ( ph  ->  ( # `  (
1 ... ( S  - 
1 ) ) )  =  ( S  - 
1 ) )
2015, 19oveq12d 6313 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  (
1 ... ( R  - 
1 ) ) )  x.  ( # `  (
1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
2111, 20syl5eq 2520 . . . . . . . 8  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  =  ( ( R  -  1 )  x.  ( S  - 
1 ) ) )
22 erdsze.n . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN )
2322nnnn0d 10864 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN0 )
24 hashfz1 12399 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2523, 24syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
269, 21, 253brtr4d 4483 . . . . . . 7  |-  ( ph  ->  ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) ) )
27 fzfid 12063 . . . . . . . 8  |-  ( ph  ->  ( 1 ... N
)  e.  Fin )
28 hashsdom 12429 . . . . . . . 8  |-  ( ( ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  e.  Fin  /\  (
1 ... N )  e. 
Fin )  ->  (
( # `  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) )  <  ( # `  (
1 ... N ) )  <-> 
( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
) ) )
294, 27, 28sylancr 663 . . . . . . 7  |-  ( ph  ->  ( ( # `  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )  <  ( # `  ( 1 ... N
) )  <->  ( (
1 ... ( R  - 
1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  ~<  (
1 ... N ) ) )
3026, 29mpbid 210 . . . . . 6  |-  ( ph  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
) )
31 erdsze.f . . . . . . . 8  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
32 erdszelem.i . . . . . . . 8  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
33 erdszelem.j . . . . . . . 8  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
34 erdszelem.t . . . . . . . 8  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
3522, 31, 32, 33, 34erdszelem9 28475 . . . . . . 7  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
36 f1f1orn 5833 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T : ( 1 ... N ) -1-1-onto-> ran  T )
37 ovex 6320 . . . . . . . 8  |-  ( 1 ... N )  e. 
_V
3837f1oen 7548 . . . . . . 7  |-  ( T : ( 1 ... N ) -1-1-onto-> ran  T  ->  (
1 ... N )  ~~  ran  T )
3935, 36, 383syl 20 . . . . . 6  |-  ( ph  ->  ( 1 ... N
)  ~~  ran  T )
40 sdomentr 7663 . . . . . 6  |-  ( ( ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ( 1 ... N
)  /\  ( 1 ... N )  ~~  ran  T )  ->  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
4130, 39, 40syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) 
~<  ran  T )
428, 41nsyl3 119 . . . 4  |-  ( ph  ->  -.  ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
43 nss 3567 . . . . 5  |-  ( -. 
ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  E. s
( s  e.  ran  T  /\  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
44 df-rex 2823 . . . . 5  |-  ( E. s  e.  ran  T  -.  s  e.  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  E. s ( s  e. 
ran  T  /\  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) ) )
4543, 44bitr4i 252 . . . 4  |-  ( -. 
ran  T  C_  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) ) )
4642, 45sylib 196 . . 3  |-  ( ph  ->  E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
47 f1fn 5788 . . . 4  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  ->  T  Fn  ( 1 ... N ) )
48 eleq1 2539 . . . . . 6  |-  ( s  =  ( T `  m )  ->  (
s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) ) )
4948notbid 294 . . . . 5  |-  ( s  =  ( T `  m )  ->  ( -.  s  e.  (
( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5049rexrn 6034 . . . 4  |-  ( T  Fn  ( 1 ... N )  ->  ( E. s  e.  ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N )  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5135, 47, 503syl 20 . . 3  |-  ( ph  ->  ( E. s  e. 
ran  T  -.  s  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N )  -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
5246, 51mpbid 210 . 2  |-  ( ph  ->  E. m  e.  ( 1 ... N )  -.  ( T `  m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  - 
1 ) ) ) )
53 fveq2 5872 . . . . . . . . . 10  |-  ( n  =  m  ->  (
I `  n )  =  ( I `  m ) )
54 fveq2 5872 . . . . . . . . . 10  |-  ( n  =  m  ->  ( J `  n )  =  ( J `  m ) )
5553, 54opeq12d 4227 . . . . . . . . 9  |-  ( n  =  m  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  m ) ,  ( J `  m ) >. )
56 opex 4717 . . . . . . . . 9  |-  <. (
I `  m ) ,  ( J `  m ) >.  e.  _V
5755, 34, 56fvmpt 5957 . . . . . . . 8  |-  ( m  e.  ( 1 ... N )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5857adantl 466 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( T `  m )  =  <. ( I `  m ) ,  ( J `  m )
>. )
5958eleq1d 2536 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  (
( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  <. ( I `
 m ) ,  ( J `  m
) >.  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) ) ) )
60 opelxp 5035 . . . . . 6  |-  ( <.
( I `  m
) ,  ( J `
 m ) >.  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <-> 
( ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
6159, 60syl6bb 261 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  (
( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( (
I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `
 m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6261notbid 294 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  -.  (
( I `  m
)  e.  ( 1 ... ( R  - 
1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
63 ianor 488 . . . 4  |-  ( -.  ( ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  /\  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) )  <->  ( -.  (
I `  m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
6462, 63syl6bb 261 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... N
) )  ->  ( -.  ( T `  m
)  e.  ( ( 1 ... ( R  -  1 ) )  X.  ( 1 ... ( S  -  1 ) ) )  <->  ( -.  ( I `  m
)  e.  ( 1 ... ( R  - 
1 ) )  \/ 
-.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6564rexbidva 2975 . 2  |-  ( ph  ->  ( E. m  e.  ( 1 ... N
)  -.  ( T `
 m )  e.  ( ( 1 ... ( R  -  1 ) )  X.  (
1 ... ( S  - 
1 ) ) )  <->  E. m  e.  (
1 ... N ) ( -.  ( I `  m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `
 m )  e.  ( 1 ... ( S  -  1 ) ) ) ) )
6652, 65mpbid 210 1  |-  ( ph  ->  E. m  e.  ( 1 ... N ) ( -.  ( I `
 m )  e.  ( 1 ... ( R  -  1 ) )  \/  -.  ( J `  m )  e.  ( 1 ... ( S  -  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2818   {crab 2821    C_ wss 3481   ~Pcpw 4016   <.cop 4039   class class class wbr 4453    |-> cmpt 4511    X. cxp 5003   `'ccnv 5004   ran crn 5006    |` cres 5007   "cima 5008    Fn wfn 5589   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6295    ~~ cen 7525    ~<_ cdom 7526    ~< csdm 7527   Fincfn 7528   supcsup 7912   RRcr 9503   1c1 9505    x. cmul 9509    < clt 9640    - cmin 9817   NNcn 10548   NN0cn0 10807   ...cfz 11684   #chash 12385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-hash 12386
This theorem is referenced by:  erdszelem11  28477
  Copyright terms: Public domain W3C validator